On the solution of linear hydrodynamic stability of dean flow by using three semi-analytical approaches

Abstract

In the present paper, three semi-analytical techniques are examined for solving eigenvalue problem arising from linear hydrodynamics stability of Dean Flow. To this accomplishment, hybrid of Fourier transform and Adomian decomposition method (FTADM), differential transform method (DTM) and Homotopy perturbation method (HPM) is selected and applied on the eigenvalue problem. Semi-analytical results are validated against the existing data with high accuracy. The comparison between FTADM, DTM and HPM reveals that for the same number of truncated terms, the accuracy of the FTADM is more pronounced. This may be attributed to the incorporation of all boundary conditions into our solution when using the FTADM. The results also indicate that the value of wave number (i.e., parameter engaged in our eigenvalue problem) is remarkably impressive on the convergence trend and effectiveness (i.e., the occurrence of becoming nearer to the numerical results) of our solution. In addition, critical wave number and Dean number for the onset of Dean flow instability are successfully reported.

It is notable that the coefficients of \( A \),\( B \),\( C \),\( K \),\( H \) and \( M \) in the 3rd part of Eq. (37) are similar to Eq. (55), and only the index of \( n \) in \( f_{n} \left( \eta \right) \) will increase one unite (i.e., \( f_{1} \left( \eta \right) \)), and so on.